Clustering Baseball Players

Please email for any questions, comments, concerns, or discussion points about the analysis presented within. For the code used for this project, please clustering_part1 found in the main repository

Abstract

The aim of this study was to identify the different hitting styles of players in the MLB. A total of 450 players were sampled from 2015-2019 given they had at least 400 total PA, 100 PA in 2018 and 125 PA in 2017 to account for the survivorship basis and issues with identifying aging player’s hitting styles. These sub-groups were determined using four different types of Cluster Analysis: K-Means, K-Medoids, Hierarchical Clustering, and Gaussian Mixture Models. Players were clustered on their BB%, K%, and Quality of Contact %, similar to Statcast’s similarity measures for an initial comparison. Upon applying the various algorithms, K-Means and K-Medoids both found 2 hitters, while Hierarchical Clustering and Gaussian Mixture Models showed 3 different types of hitters. These types of hitters were labeled as Power-Hitters and Table Setters, with the third group as Platoon in the case of Hierarchical Clustering and Gaussian Mixture Models. Scouts and analysts can utilize them to determine a hitter’s profile objectively and find suitable replacements for batting roles for upcoming trades.

Introduction

Over the years, I have listened to many baseball commentators talk about hitters as they come up to the plate. With statements such as ‘He is a pull hitter.’, ‘He hits for power!’, ‘Gets on base a lot.’, ‘Rarely strikeouts, puts the ball in play’, a commentator utilizes a player’s profile to help guide fans. On the other hand, scouts tend to look for five qualities in players: Running Speed, Throwing Strength, Defensive skills, Hitting for Average, and Hitting for Power. Thus, a logical question is - what qualities determine these types of hitters and who else is comparable to them?

If you think this question sounds similar to Statcast’s similarity measures, it is. In my case, I want to find different groups or styles of players, whereas Statcast is interested in finding how similar players are to one another. Although analyzing similarities between two players is good, we still do not have a way of contextualizing players. For example, according to Statcast, Mike Trout’s most similar player (in 2019) is Yordan Alvarez whereas DJ LeMahieu’s is Robinson Cano. Does that mean the Yankees in 2019 would want Cano hitting where LeMahieu was or the Angels should look at Alvarez more closely if they ever need a replacement batter? Potentially. But wouldn’t a better question be what type of Hitting Style do Trout and LeMahieu posses? If a team needs someone to hit for power, who should they look to acquire? Similarity measures cannot fully answer either of these questions. It can tell a team who would be the most similar replacement, but not what exactly are they trying to replace or acquire. Additionally, just because a player is a suitable replacement, does not mean they would be the best for the job. A team looking to replace an average player may replace them with another average player, but would it not be more beneficial to replace them with a player that fits the desired role more closely and ideally cheaper?

Thus, instead of analyzing how similar two players are, we should attempt to group them. In order to do this, I will use a method called Cluster Analysis (CA), an unsupervised learning method, where researchers attempt to find similar groups between observations.

Data

An original total of 548 players were sampled given they had a minimum of 400 PA between 2015-2019. This dataset was further refined to remove players who had less than 100 PA in 2018 and less than 125 PA in 2017. This reduction process was implemented to account for any survivorship basis and player’s missing time due to injuries. Some additional players were included such as Miguel Andujar, Travis Jankowski, Preston Tucker, Ryan Flaherty, Bobby Wilson, Peter Bourjos, and Rene Rivera. Additionally, some players did retire in the timeframe like A.J. Ellis, but were included as there was no efficient way to check them other than a manual search. While this favors players who have played more recently, given Statcast’s Expected Stats and Quality of Contact only goes back to 2015, I did not think it correct to evaluate an aging player’s last few years. I could have decomposed each year of a player’s performance, but Statcast does not like it when you load a lot of data. Thus, the total sample size of players was 450.

For part 1 of this analysis, I clustered players on their BB%, K%, and Quality of Contact rates using a few different methods. In the future, I plan on incorporating more variables such as direction and production. All data was compiled with Statcast’s search function and FanGraphs, which are both free to use.

Clustering Methods

There are many ways to cluster observations, but I think the most popular methods are K-Means, K-Medoids, Hierarchical Methods with dendrograms, and Gaussian Mixture Models. K-Means and K-Medoids partitions n observations into k clusters, where each observation belongs to the cluster with the nearest mean/median. A dendrogram is a tree diagram that shows a hierarchical relationship between objects. From this hierarchical relationship, we can determine the clusters by analyzing which players group together under a hierarchy. I think whenever we classify anything, one observation will probably be a mixture of the groups. For example, although Trout is in his own world according to Statcast, I suspect there exists players who hit for power, get on base, put the ball in play, etc. Each player has some of those qualities, hence Gaussian Mixture Models (GMM). GMM are probabilistic models that are able to represent sub-groups within a population; however, they also provide a probability an observation is from a sub-group. Therefore, in my opinon, they can better answer the question of how to contextualize players.

For those interested in the details for each method, the following section will be beneficial. For those looking for the results, feel free to skip to the Results section.

K-Means and K-Medoids

K-means assigns observations to clusters to minimize the within-cluster variances, which are the squared Euclidean distances. First, we specify the number of clusters, then randomly select a number of observations as the initial cluster centers. Next, we assign each observation to a cluster based on the Euclidean distance between the observation and the centroid and update the cluster centroid with the new mean. Lastly, we iterate and assign each observation to a cluster, updating the cluster centers until we have minimized the total within-cluster variances. The more general method is K-Medoids, which minimizes the sum of dissimilarities between points labeled to be in a cluster and a point designated as the center of that cluster in the same general process. K-Medoids tends to be non-sensitive to outliers and reduces noise versus K-Means because it uses the median rather than the mean. Furthermore, in contrast to the K-means algorithm, K-Medoids chooses datapoints as centers.

For K-means, I leveraged R’s NbClust package, which evaluates a different number of clusters over 26 different cluster indices and picks the optimal classification based on the majority rule. I evaluated 2-30 clusters. This is primarily because people agree that there are different types of hitting styles, but do not agree on the exact number. For example, Gary Huckabay believes there are 26 distinct aging curves each applying to a different type of player. This prediction of 26, which comes with a lot of experience and art, is still subjective.

For K-Medoids, I implemented it with the Partitioning Around Medoids (PAM) algorithm and the number of clusters was evaluated using the Silhouette Index as is best practice. Instead of using Euclidean distance, I used the Minkowski distance with p = 0.34. The Euclidean distance is actually a generalized version of the Minkowski distance with p = 2 for those that know the equation. “Why 0.34?”, do you ask, well that specific value provided the maximum average silhouette value for my clusters in testing.

Hierarchical Methods

Hierarchical Clustering builds tree diagrams that show some hierarchy. Similar to an organizational charts with a CEO at the top and the different departments below them. The approach to creating these tree diagrams, or dendrograms, are either top-down, “Divisive”, or bottom-up, “Agglomerative”. In top-down, we start with 1 cluster and try to split the observations as we move down, while bottom-up is the reverse, everybody is in their own cluster and we move up the tree grouping observations. For Hierarchical Clustering (HC), there are two main parameters to adjust: the distance metric and the linkage method. The distance method will measure how far or close the points are in some n dimensional space and the linkage method describes the process in which we form the clusters. For simplicity, I used the Euclidean distance and the Complete linkage, an agglomerative method.

Distance metrics are extremely important for our data and some linkage methods require a certain distance in order to be “correct”. Also, the choice of distance metric can be fairly important from a data perspective. Consider the case of clustering shoppers based on their past shopping history in terms of the number of each good they bought. If we used Euclidean distance, shoppers that did not buy a lot of items would be clustered together and shoppers that did buy a lot would be in another cluster. But instead, if we used a correlation-based distance metric, then shoppers with similar preferences would be grouped together. In this case, we are analyzing players based on their percentages to control for playing time, we do not need to use a correlation-based distance metric. If we analyzed the raw Ks or BBs, as in how many Ks or BBs a player had in their career, then we would want to use correlation-based distance metrics.

There are 4 main linkage methods: Complete, Single (Nearest Neighbor), Average, and Centroid. Complete and Average tend to produce more balanced dendrograms and they seek to first compute the pairwise dissimilarities between two observations, then record the largest (Complete) or average (Average) of these dissimilarities. Single linkage is the opposite of Average and Complete, it combines the closest pair of elements not yet belonging to the same cluster. Lastly, Centroid linkage tends to be used in genomics.

Therefore, in order to determine the number of clusters for Hierarchical Clustering, I leveraged NbClust once again, but this time using Complete Linkage as the method instead of K-means.

Gaussian Mixture Models

GMM are a mixture of Gaussian probability distributions. A Gaussian distribution is essentially a bell curve with a certain mean and variance. The probability density function for the mixture is a linear combination of these individual distributions. For example, in Figure 1, we have two Gaussian distributions, x1 and x2 in orange and their mixture model in blue. In 2 dimensions, with different variables, we have something that looks similar to the contour lines on a map, Figure 2.

GMM assume there are a certain number of Gaussian distributions, which each representing a cluster, with corresponding mean and covariance. Formally, for a given set of data points, X, we have a mixture of k Gaussians; however, the question becomes, how do we determine the mean and covariance, the parameters, for each Gaussian? In order to find these parameters, we seek to maximize the likelihood that X is most probable. In layman’s terms, we find the probability density function that best fits the dataset. In order to do this, we use the Expectation-Maximization (EM) algorithm.

The EM algorithm in GMM is an iterative process. First, we assume we have k clusters. Then we need to calculate the parameters for each of these clusters, remember they are Gaussian distributions, as follows:

  • Expectation Step: First, for each observation, xi, we calculate the probability it belongs to some cluster c1, c2, …, ck.
  • Maximization Step: Based on the probability, we then update the parameters for the mean, covariance, and density of each cluster.

This process repeats until we have maximized the likelihood.

Additionally, each cluster has certain geometric features, such as the shape/spread of the density contours, volume of the cluster (number of objects in a cluster), and orientation, which are determined by their covariance matrix and a distribution structure type, such as spherical, diagonal, or ellipsoidal. These distributions can be described and visualized in the following ways:

  • Spherical: circular contour lines, cone like structure
  • Diagonal: circular contour lines but stretched in a coordinate direction, x or y
  • Full/Ellipsoidal: circular contour lines but stretched in multiple directions

For the Spherical distribution, there are 2 models because they can have either equal or variable volume. Their shape is the same in either direction and the orientation is indifferent because they are a circle, a circle rotated on a 2D plane results in the same circle. There are 4 models for diagonal distributions, they each can have equal or variable volume and shape orientated on the coordinate axes. Lastly, there are 8 models for the ellipsoidal case, and the clusters can have equal or variable volume, shape, and/or orientation.

To implement GMM, I used mclust in R, as we can test a range of possible clusters and compare them with the Bayesian Information Criterion, BIC. The main benefit here is when fitting models, it is possible to increase the likelihood by adding more parameters, but this may result in overfitting. Thus, BIC introduces a penalty term for the number of parameters and when measuring, the lower the BIC, the more optimal the model. That said, in mclust, BIC is calculated in reverse because the number of clusters is not considered an independent parameter for calculating BIC. Thus, we are maximizing the BIC when using this package. I am going to skip the details as to what exactly is going on under the hood here because they are not totally important in this study, but a good explanation can be found on Cross-Validated. This type of process, maximizing a value when we generally minimize it, is also found in other processes like sklearn’s cross validation package with regression and Mean Square Error, for those curious.

Results

K-Means and K-Medoids

When using K-means and NbClust, 2 clusters were determined to be the optimal number based on the majority rule. Similarly, when implementing K-Medoids, 2 clusters were also determined when using the Silhouette Index as the validation index.

Hierarchical Analysis with Dendograms

When clustering using Complete linkage and Euclidean distance, 13 indexes determined 3 clusters as the optimal classification and presented the following structure.

Gaussian Mixture Models

Upon implementing GMM, mclust tested each model using a range of 1-10 clusters. From this analysis, the optimal number of clusters was 3 and they were Ellipsoidal with variable volume, shape and equal orientation, VVE.

Visualizing and Discussing Clusters

With these clustering objects, we can visualize the clusters in their 2D space and their statistical profile using radar charts. We already visualized the results of the HC method above as the dendrogram.

Most noticeabe is the amount of variance that is explained in the dimensions. Using Euclidean distance in K-Means and GMM, we only explained 56% of the data in 2D, whereas when we changed to the Minkowski distance, we were able to explain 62% of the variance. Overall this is not a tremendous amount within our dataset, and some believe it should be at least 60%. In order to look at the variables that accounted for the most variance, we can look at a biplot of the variables, using Euclidean distance. Here we see that Weak% and Flare% do not contribute all that much to the underlying clusters, as in they are not good differentiators, and may be potential variables to remove.

For those that are observant, you can see that K-Means and GMM explain the same amount of total variance in 2D, with K-Medoids explaining a little more. K-Medoids is higher because we used a different distance metric. GMM is the same because K-means is in fact a special case of EM for GMM that assumes a hard assignment (no probabilities) and the identity matrix is used for the covariance. Thus, we are operating in the same dimensional space for both GMM and K-Means.

In analyzing the 2D plots, the question are: Does the dataset contain meaningful clusters, as in a non-random structure or not? We call this assessing cluster tendency. To test, I used the Hopkins statistic. In implementing this test, we are testing the alternative hypothesis that the dataset is not uniformly distributed and if H < 0.5, then it is unlikely we have statistically significant clusters. Thankfully, when I ran it on this dataset, I had H = 0.73, thus we have clusterable data.

With these clusters, it always useful to label them and show their centers. For K-Means and K-Medoids, I decided to call the two clusters Power-Hitters and Table-Setters. However, in HC and GMM Clustering, where we found 3 clusters, I called the third cluster Platoon players. From a profile view, Power-Hitters are fairly straightforward. Those players have lots of Barrels and Solid contact rates, they tend to Strikeout and Walk more. On the other hand, the Table-Setters tend to do the opposite, they hit more Topped balls. Therefore, they do not have the power to turn those Topped balls into Solids or Barrels. They may get on base, but tend to hit for singles rather than extra bases I suspect.

For the Platoon players, they fit somewhere in the middle of that. In the case of HC, the main difference is that they Walk the same amount as Table-Setters, but get the same number of Strikeouts, Barrels, and Topped. For GMM, accounting for the variance between the variables, these Platoon players perform very differently than their cohort in HC. They do not Barrel the ball as much as the Table-Setters, but they Strikeout at a similar rate as Power-Hitters and have a similar number of Topped balls and Walks as Table-Setters. These are the players, to me, tend to not play frequently because they do not offer the same level of consistency as the other two groups. They may suffer from not having the same amount of playing time, thus there is probably more variance within this group.

Probably the question that people care about is: Who are in these groups? The full output is in the Cluster Results section of this repository. The results can be controversial, and the names themselves are subjective The names were informed by the radar charts, not from the players within them. Honestly, I am happy with the names for K-Means, K-Medoids, and GMM, but not the HC method. HC had a different number of observations per cluster versus all the other methods. Additionally, the names within the HC groups did not always make sense. For example, Mookie Betts and Manny Machado were both classified as Platoon players, but in the three other methods they were Table-Setters or at least a mix between Table-Setters and Power-Hitters, never Platoon. That does not mean this analysis is incorrect, Cluster Analysis is subjective, but not even I like HC. I called them Platoon because we are not looking at defensive metrics, thus it would be inadequate to call them Defensive Specialists. If somebody has a better name for them, in the HC case, please by all means suggest it.

For simplicity, I will only discuss some of the GMM results within here. I am happy to discuss all the results through email and I have posted my contact information below. In terms of a Top 5 from each category, both the Power-Hitters and Platoon players had players in the Top 5 with 100% classification, meaning they are only this type of player. To no surprise, Joey Gallo, Mike Trout, Aaron Judge, Giancarlo Stanton are the top 5 Power Hitters, with Jake Cave rounding it out. For the Platoon players, Billy Hamilton, Bradley Zimmer, Travis Jankowski, Delino Deshields, and Jarrod Dyson are the Top 5. For the Table-Setters, none of them are 100%, but they are all above 99.5%. These players are Daniel Murphy, Jeff McNeil, Yuli Gurriel, Michael Brantley, and Yadier Molina.

However, the best part of Gaussian Mixture Models is the probabilities. For example, Brian Goodwin was a 39%/35%/26% player in terms of Table-Setter%/Power-Hitter%/Platoon%. There is ambiguity about his style of hitting. This is a good thing! Yes, we all agree that Trout, Judge, Stanton, etc. are Power-Hitters. But can someone honestly tell me what type of hitter Brian Goodwin is? If we used K-Means or HC, we would have said he is a Power Hitter or Platoon respectively, but there is a good portion of him that could also be a Table-Setter. My point is, when we are doing some subjective analysis like clustering, the results need to have probabilities attached to them.

Conclusions

Hopefully, this work shows first and foremost there are other methods besides K-Means that work and, in my opinion, are much better, as they leverage both mean and the covariance. Too frequently, I see budding analysts apply K-Means by picking a subjective number of clusters and proceeding with their analysis. It would be beneficial to think more in terms of probabilities.

Finally, the main reason we are all here: Are there two, three, or 26 different types of hitters? Based on my analysis, I would argue there are three different types of hitters. We could assume there are two, but I prefer to think of my results also in terms of a batting-order. For spots 1 and 2, a team tends to have Table-Setters. For spots 3-6, Power-Hitters. Then 7, 8, and 9, tend to be either the pitcher in the NL non-COVID season, your more Platoon and Defensive players, or someone who is very fast. Having only two different types of hitters is too simplistic, whereas three types of hitters is something that we see every day. There might have been a time when we had 26 different types of hitters, but over time with teams focusing on analytics, many types of hitters have fallen out of favor. Also, Huckabay argued there were 26 different aging curves, but he did not specify if they were hitting aging curves. Therefore, we have Power-Hitters, Table-Setters, and Platoon Hitters in the game today.

In the Future…

For Part 2, I would like to incorporate more variables such as Pull%, Center/Straightaway%, Opposite%, and also measures of production such as xBA, xISO, xwOBA. This way we could know more objectively the difference between a Table-Setter and a Power-Hitter. I do believe that a combination of these variables will provide “better” groupings and explain more the of the dataset rather than 56%.

Additionally, I am going to take a deeper dive using dimensionality reduction techniques to analyze the variables more thoroughly. As we saw in the radar charts and biplot, Weak% and Flare% did not provide a good differentiator. Every group had roughly the same percentage.

In terms of clustering methods, I am leaning towards only focusing on GMM. I prefer soft-clustering methods because we can have conversations about players that can be a combination of all groups. Hard-clustering methods state an observation is in this cluster, hence stating a player is only this type of player. Although that may be useful for commentators that are hedgehogs, I prefer to be a fox (see Nate Silver’s hedgehog vs fox analogy). Players can have aspects from multiple groups.

Comments and Questions?

Please email for any questions, comments, concerns, or discussion points about the analysis.